An analytical model of a curved beam with a T shaped cross section

• Published in: Journal of sound and vibration 2018-03, Vol.416, p.29-54
• Main Authors: Hull, Andrew J., Perez, Daniel, Cox, Donald L.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Continuity equation; Cross-sections; Curved beam; Curved beams; Differential equations; Dynamic models; Elasticity; Finite element analysis; Finite element method; Propagation; Ring vibrations; Studies; T beams; T section; Two dimensional models; Wave propagation; Webs
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2017.11.044

This paper derives a comprehensive analytical dynamic model of a closed circular beam that has a T shaped cross section. The new model includes in-plane and out-of-plane vibrations derived using continuous media expressions which produces results that have a valid frequency range above those available from traditional lumped parameter models. The web is modeled using two-dimensional elasticity equations for in-plane motion and the classical flexural plate equation for out-of-plane motion. The flange is modeled using two sets of Donnell shell equations: one for the left side of the flange and one for the right side of the flange. The governing differential equations are solved with unknown wave propagation coefficients multiplied by spatial domain and time domain functions which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to results from finite element analysis. •Analytical model of a ring with a T shaped cross-sectional area is derived.•The displacement field is formulated using nine separate differential equations.•The displacement field is solved using equilibrium and continuity equations.•A numerical example is included and compared to finite element results.